Fixed points of coisotropic subgroups of $\Gamma_{k}$ on decomposition spaces

Gregory Arone ; Kathryn Lesh

Homology, homotopy, and applications, Tome 22 (2020) no. 1, pp. 77-96.

Résumé

We study the equivariant homotopy type of the poset ${\mathcal{L}}_{p^k}$ of orthogonal decompositions of ${\mathbb{C}}^{p^k}$. The fixed point space of the $p$-radical subgroup $\Gamma_{k}\subset U(p^k)$ acting on ${\mathcal{L}}_{p^k}$ is shown to be homeomorphic to a symplectic Tits building, a wedge of $(k-1)$-dimensional spheres. Our second result concerns $\Delta_{k}=({\mathbb{Z}}/p)^{k}\subset U(p^k)$ acting on ${\mathbb{C}}^{p^k}$ by the regular representation. We identify a retract of the fixed point space of $\Delta_{k}$ acting on ${\mathcal{L}}_{p^k}$. This retract has the homotopy type of the unreduced suspension of the Tits building for $\operatorname{GL}_{k}\!\left({\mathbb{F}}_{p}\right)$, also a wedge of $(k-1)$-dimensional spheres. As a consequence of these results, we find that the fixed point space of any coisotropic subgroup of $\Gamma_{k}$ contains, as a retract, a wedge of $(k-1)$-dimensional spheres. We make a conjecture about the full homotopy type of the fixed point space of $\Delta_{k}$ acting on ${\mathcal{L}}_{p^k}$, based on a more general branching conjecture, and we show that the conjecture is consistent with our results.

Détail

DOI : 10.4310/HHA.2020.v22.n1.a6

Classification : 55N91, 55P65, 55R45
Keywords: Tits building, decomposition space, fixed points, unitary group